A plane accelerates from rest at a constant rate of 6.00  m/s2
  along a runway that is 1860  m
  long. Assume that the plane reaches the required takeoff velocity at the end of the runway. What percentage of the takeoff velocity did the plane gain when it reached the midpoint of the runway?


We can solve this problem using the kinematic equations for constant acceleration. Let’s go step by step.

### Key Information:
- Constant acceleration, \( a = 6.00 \, \text{m/s}^2 \)
- Total runway length, \( d = 1860 \, \text{m} \)
- Initial velocity, \( u = 0 \, \text{m/s} \) (since the plane starts from rest)

We are tasked with finding what percentage of the final velocity \( v_f \) the plane has gained at the midpoint of the runway.

### Step 1: Calculate the final velocity \( v_f \) at the end of the runway.

We will use the kinematic equation that relates acceleration, distance, and velocity:

\[
v_f^2 = u^2 + 2ad
\]

Since the plane starts from rest, \( u = 0 \), the equation simplifies to:

\[
v_f^2 = 2ad
\]

Substitute the known values:

\[
v_f^2 = 2(6.00)(1860)
\]
\[
v_f^2 = 22320 \, \text{m}^2/\text{s}^2
\]
\[
v_f = \sqrt{22320} \approx 149.40 \, \text{m/s}
\]

Thus, the takeoff velocity \( v_f = 149.40 \, \text{m/s} \).

### Step 2: Find the velocity \( v_m \) at the midpoint of the runway.

The midpoint of the runway corresponds to half the distance, i.e., \( d_m = \frac{1860}{2} = 930 \, \text{m} \).

We use the same kinematic equation to find the velocity at the midpoint:

\[
v_m^2 = u^2 + 2ad_m
\]

Again, since \( u = 0 \):

\[
v_m^2 = 2(6.00)(930)
\]
\[
v_m^2 = 11160 \, \text{m}^2/\text{s}^2
\]
\[
v_m = \sqrt{11160} \approx 105.64 \, \text{m/s}
\]

### Step 3: Calculate the percentage of the final velocity gained at the midpoint.

The percentage can be calculated as:

\[
\text{Percentage} = \left( \frac{v_m}{v_f} \right) \times 100
\]

Substitute the values:

\[
\text{Percentage} = \left( \frac{105.64}{149.40} \right) \times 100 \approx 70.7\%
\]

### Final Answer:
The plane gained approximately **70.7%** of its takeoff velocity when it reached the midpoint of the runway.

---

Do you want further details or have any questions?

Here are five related questions to expand on this problem:
1. How much time does it take for the plane to reach takeoff velocity?
2. What is the acceleration if the plane takes 30 seconds to reach takeoff velocity on a different runway?
3. What would happen if the runway was only 1000 meters long?
4. How does the required takeoff velocity change with varying runway lengths?
5. How do external factors like wind resistance affect this calculation?

**Tip:** When dealing with kinematics problems, always start by identifying the known variables and the correct equation that ties them together.

A plane accelerates from rest at a constant rate of 6.00 m/s² along a runway that is 1860 m long. Assume that the plane reaches the required takeoff velocity at the end of the runway. What percentage of the takeoff velocity did the plane gain when it reached the midpoint of the runway?

Solve a kinematics problem involving a plane accelerating at 6.00 m/s² along a 1860 m runway. Learn how to calculate what percentage of the takeoff velocity is gained at the midpoint using constant acceleration equations. Suitable for Grades 10-12 students studying physics.

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